# Single point Seshadri constants on rational surfaces

## TL;DR

This paper constructs examples of irrational single-point Seshadri constants on rational surfaces obtained by blowing up very general points in the complex projective plane, under a mild geometric assumption related to negative curves.

## Contribution

It demonstrates irrational Seshadri constants on rational surfaces assuming only that negative self-intersection divisors are smooth rational curves, a weaker condition than the full SHGH Conjecture.

## Key findings

- Existence of irrational Seshadri constants on rational surfaces.
- Relates negative curves to Seshadri constant irrationality.
- Provides evidence supporting conjectures on Seshadri constants.

## Abstract

Motivated by a similar result of Dumnicki, K\"uronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface $X$ obtained by blowing up very general points of $\mathbb{P}^2_\mathbb{C}$, assuming only that all prime divisors on $X$ of negative self-intersection are smooth rational curves $C$ with $C^2=-1$. (This assumption is a consequence of the SHGH Conjecture, but it is weaker than assuming the full conjecture.)